Mathematical physics·OpenAI Math·Result 278
Failure of Kohn–Sham ensemble representation
One of the 372 results in OpenAI's public mathematics catalogue, filed under mathematical physics: 1 preprint, the latest dated 25 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports a result titled ‘Failure of Kohn–Sham ensemble representation’ (mathematical physics) in a preprint dated 25 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More, is on YouTube.
Chapter 81 of 84 in Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · starts at 27:46 · watch from there on YouTube →
What OpenAI claims
Constructs a three-electron Coulomb molecule with two equal positive-integer-charge nuclei whose absolute ground-state density has no noninteracting ground-state ensemble representation by a single real spin-independent local potential in L3/2(ℝ3)+L∞(ℝ3). This disproves Kohn–Sham ensemble representability for that potential class; the required nuclear charge is specified nonnumerically.
Paper
A Coulomb ground-state density without Kohn-Sham ensemble representation
OpenAI's abstract We construct a finite three-electron Coulomb molecule whose spin-summed ground-state density cannot be reproduced by any ground-state ensemble of noninteracting electrons in a single real, spin-independent local potential in L3/2(R3)+L∞(R3). The molecule has two equal positive integer nuclear charges, specified by an exact finite nonnumerical formula.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 25 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- Constructs a three-electron Coulomb molecule with two equal positive-integer-charge nuclei whose absolute ground-state density has no noninteracting ground-state ensemble representation by a single real spin-independent local potential in L3/2(ℝ3)+L∞(ℝ3). This disproves Kohn–Sham ensemble representability for that potential class; the required nuclear charge is specified nonnumerically.
- Where is the proof?
- In the paper, A Coulomb ground-state density without Kohn-Sham ensemble representation (PDF).
- Is there a video?
- Yes, on YouTube (29:10, chapter 81).
Also in
The video
Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · Compilation · 29:10 · chapter 81 from 27:46 · description, sources and chapters →
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